*Why* do you (think you) need to give the end user
a caret in a non-editable text area?

Since the caret indicates the place that has the *input*
focus, and there is no input for a non editable component,
it does not make that much sense.  (And would be confusing).

Note that in this example, the text *selection* shows clearly,
that is probably all the user needs.

</sscce>
import java.awt.*;
import javax.swing.*;

public class CaretInText extends JPanel {

 CaretInText() {
   setLayout( new BorderLayout() );

   JTextPane tp = new JTextPane();
   tp.replaceSelection("Hi There!");
   tp.setCaretPosition(0);
   tp.moveCaretPosition(3);
   tp.setEditable(false);

   add( tp, BorderLayout.CENTER );
 }

 public static void main(String[] args) {
   JFrame f = new JFrame("Caret In Text");
   f.getContentPane().add( new CaretInText() );
   f.setDefaultCloseOperation( JFrame.DISPOSE_ON_CLOSE );
   f.pack();
   f.setVisible(true);
 }
}
</sscce>

HTH